To test for the presence of a warning signal in these indicators we
computed values of Kendall's \(\tau \)  for both indicators for each of
the 266 replicates. Kendall's \(\tau \)  is a non-parametric measure
of rank correlation frequently used to identify an increasing trend
(\(\tau > 0 )\)  in early warning signals [1]}, [2]},
defined as \(\tau {\frac{1}{2} n (n-1) }\)  in \(n\) 
observations. A pair of observations \((x_i,y_i)\)  and \((x_j,y_j)\)  are concordant if \(x_i > x_j\)  and \(y_i > y_j\)  or \(x_i < x_j\)  and
\(y_i < y_j\)  and discordant otherwise; equalities excepted. \(\tau \)  takes
values in \((-1, 1)\) . The distribution of \(\tau \)  values observed across
these replicates is shown in Figure REF . We compare
the distribution of \(\tau \)  from all the simulations to the distribution
conditioned on experiencing a chance transition to the alternative
stable state. To avoid an effect of sample size the time series are
all chosen to be the same length.
